1. Economic Capital and the Aggregation of Risks Using Copulas
Good morning everyone, I am Andrew Tang and today I am presenting the paper titled “Economic Capital and the Aggregation of Risks Using Copulas” by Associate Professor Emil Valdez and myself.
Essentially this paper looks at using copulas to model dependencies between losses from different lines of business for a general insurance company and in particular, the capital requirements for insurers that are implied by these models.
Economic Capital and the Aggregation of Risks Using Copulas
Dr. Emiliano A. Valdez and Andrew Tang

2. Overview Motivation and aims Technical background - copulas
Numerical simulation
Results of simulation
Key findings and conclusions
To begin, I will first provide the underlying motivation and outline the specific aims covered in our paper.
Then I will run through some brief summary of copulas, focusing mainly on the models used in our paper.
Our analysis essentially was done through a simulation of prospective 1 year losses for a multi-line insurer, so I will also spend some time going through the details of this simulation.
Then we will have a look at some of the significant results from the simulation and its implications for capital requirements, before finishing up with a summary of the key findings and what we can, and cannot conclude from our work.

3. Capital
Buffer
A rainy day fund, so when bad things happen, there is money to cover it
Quoted from the IAA Solvency Working Party (2004) – “A Global Framework for Solvency Assessment”
Solvency and financial strength indicator
Economic capital - worst tolerable value of the risk portfolio
Generally, to operate any insurance business, we need a certain amount of capital. In the most basic sense, this is the excess of the insurer’s assets over its liabilities.
An insurer needs capital as a buffer, or cushion, so that it can continue to pay out claims and write new business under adverse financial conditions. The IAA Solvency Working Party gives capital the definition of “a rainy day fund, so when bad things happens, there is money to cover it.” This implies basically that capital is needed to ensure the ongoing operation of the company and to protect policyholders’ claims on the company.
Apart from ensuring solvency, capital also serves as an indicator of the financial strength of an insurer. Therefore, depending on what the context is, we may talk about many different types of capital such as regulatory capital and rating agency’s capital. In our paper, we focus on what is known as economic capital, which represents the amount required to be held given an insurer’s risk portfolio. We are all aware that it isn’t possible to hold enough capital to guarantee solvency, therefore, we define economic capital in our paper as the worst tolerable value of the random loss. That is, in the worst tolerable situation, we must hold assets that is equal to the amount of economic capital.

4. Multi-Line Insurers Increasingly prominent
Diverse range insurance products
Aggregate loss, Z
Where Xi represents the loss variable from line i.
Xis are dependent
Over the last decade or so, we have seen consolidation in the insurance industry and as a result, multi-line insurers are becoming increasingly prominent.
These operations often write a diverse range of insurance products, including both personal and commercial lines, and property as well as liability lines of business.
In general, we denote the aggregate loss random variable as Z and this is equal to the sum of the losses generated from each business line, the X’s. Here we assume that the X’s are dependent in some sense. This dependency between business lines is widely recognised in the industry and we can readily find intuitive reason for it. For example, we can consider a faulty brake in a car causing an accident and hence triggering claims in both the motor and products liability lines of business.

5. Multi-Line Insurers Dependencies between Xis ignored
E.g., APRA Prescribed Method
Dependencies modelled using linear correlations
Inadequate
Non-linear dependence
Tail dependence
In terms of modelling loss distributions, dependencies between risks play a central role.
However, for capital calculations, we find that dependencies are not accurately factored in. They are often ignored, such as the case of APRA’s prescribed method, or are often modelled using linear correlations. Using linear correlations alone is often inadequate to capture all the dependence structure. Important aspects of the dependencies such as non-linear dependence and in particular, tail dependence, which has profound capital implications cannot be modelled using linear correlations.
In our paper, we aggregate each individual line’s losses assuming by different copulas to model the dependence structure. In fact, copulas can be shown to capture ALL aspects of the dependence structure.

6. Multi-Line Insurers Capital risk measures Capital requirements
Value-at-Risk (VaR) – quantile risk measure
Tail conditional expectation (TCE)
To measure the capital requirement, we rely on using risk measures, which basically quantifies the degree of risk within a portfolio, and hence the required capital.
A risk measure, rho, is simply a mapping of the risk – the aggregate loss, onto the set of real numbers. Suppose we call the capital requirement as K, then in our paper, K simply takes the value of the risk measure, rho(Z).
In our paper, we use two widely known risk measures to measure capital, the value at risk, or quantile measure and the tail conditional expectation measure. We also proved in our paper that based on minimising the cost of insolvency, these risk measures are in fact optimal for measuring capital.

7. Multi-Line Insurers Diversification benefit q = 97.5% and 99.5%
If we have the distribution of the aggregate loss, Z, as well as the the distributions of the individual line’s losses Xi’s, we can show that in general, the capital required for the aggregate loss is smaller than or equal to the sum of the capital required for each individual line’s loss. This is analogous to saying that a multi-line insurer will always require less capital than if it were to operate its business as separate lines.
We call this difference the diversification benefit, and apart from investigating the effect of copulas on the capital requirements, we also look at how copulas affect this diversification benefit.
For our paper, we have chosen our quantiles to be 97.5% and 99.5% in determining the capital requirements. These values were chosen arbitrarily but the 99.5% value was chosen specifically so that we can compare our results from the copula models to APRA’s Prescribed method.

8. Aims
Study the capital requirements (CRs) under different copula aggregation models
Study the diversification benefits (DBs) under different copula aggregation models
Compare the CRs from copula models to the Prescribed Method (PM) used by APRA
So to summarise, these are the three aims we want to achieve in this paper:
To study the capital requirements under different copula models
To study the diversification benefits under different copula models
And to compare the capital requirements under the different copula models and APRA’s prescribed method which is currently used by all insurers in Australia.

9. Copulas Individual line losses - X1, X2, …, Xn
Joint distribution is F(x1,x2,…,xn)
Marginal distributions are F1(x1), F2(x2), …, Fn(xn)
A copula, C, is a function that links, or couples the marginals to the joint distribution
Sklar (1959)
Before turning to the simulation of the actual loss variables and the calculation of the capital requirements, it is fitting to give a technical overview of copulas.
Suppose we have random losses Xi’s from each line of business, with joint distribution F(x1,x2…xn), and marginal distributions F1(x1), F2(x2) etc.
A copula, C, is simply a function that links or couples the marginals to the joint distribution, as shown here. This result was first proposed by Sklar, and he proved that for absolutely continuous marginals, there is always a unique copula representation of the joint distribution.
Therefore the given the marginal distributions of each line’s losses, and the copula function, we can simulate the full distribution of dependent losses.

10. Copulas Copulas of extreme dependence Independence copula
Archimedean copulas
Gumbel-Hougaard copula
Frank copula
Cook-Johnson copula
Now we will run through some examples of copulas.
We begin with the independence copula, which simply represents the case where the marginals are independent of one another. Hence, their joint distribution is simply the product of the marginals.
Archimedean copulas, including the Gumbel, Frank and Cook-Johnson copula, is another important class. However, in generating variables under these copulas, we are only allowed to input one parameter. But this is inadequate if we wanted to model the dependence structure for more than 2 lines of business as the single parameter will not capture all the pair-wise dependence. Therefore, these are not suitable for our present purpose.

11. Copulas Elliptical copulas / variants of the student-t copula
Gaussian “Normal” copula (infinite df)
Student-t copula (3 & 10 df)
Cauchy copula (1 df)
Where Tv(.) and tv(.) denote the multivariate and univariate Student-t distribution with v degrees of freedom respectively.
We focus on Elliptical copulas, which allows for pair-wise dependence to be inputted through a linear correlation matrix, and hence provides more flexibility when modelling more than 2 lines of business.
We use the Normal, Student-t and Cauchy copulas in this paper. In fact, all three of these examples are variants of the Student-t copula with different degrees of freedom. In the Normal case, we have an infinite number of degrees of freedom, and the Cauchy has 1 degree of freedom. We also examine the Student t-copula with 3 and 10 degrees of freedom.

12. Copulas Tail dependence (Student-t copulas)
where t* denotes the survivorship function of the Student-t distribution with n degrees of freedom.
n\r
0.5
0.9 1
0.29
0.78 3
0.12
0.31
0.67 10
0.01
0.08
0.46
infinity
One particular aspect of copulas is its ability to incorporate the tail dependence structure between random variables. This is a important feature for capital purposes as capital is really concerned about the positive tail of the aggregate loss distribution.
Here we give the tail dependence measure, lambda, for Student-t copulas. N here represents the degrees of freedom and rho represents the level of linear correlation. We clearly see that the higher the correlation, the higher the tail dependence, which is intuitive. Also, we find that lambda decreases as the degrees of freedom increases. This leads us to conclude that out of the copulas we have chosen, the Cauchy will have the heaviest tail dependence, followed by the Student-t (3 df), Student-t ( 10 df) and then the Normal copula. It should be noted that the normal copula doesn’t allow for tail dependence unless if the variables have perfect linear correlation.

13. Numerical Simulation
1 year prospective gross loss ratios for each line of business
Industry data between 1992 and 2002
Semi-annual
SAS/IML (Interactive Matrix Language)
Let’s now look at how we generated the losses from each line of business under each copula.
In our simulation, we assumed the prospective 1 year loss random variables for each line of business are represented by the gross loss ratios. This is calculated as the gross incurred claims divided by the gross earned premiums in the corresponding period. In other words, the loss variables that we simulate are really standardised claims measures, in this case, standardised by the corresponding earned premium.
All of our inputs for the simulation were derived from industry data, from APRA and the former insurance commissioner publications from 1992 to 2002.
And the actual simulation was done using a SAS program.

14. Numerical Simulation Five lines of business
Motor: domestic & commercial
Household: buildings & contents
Fire & ISR
Liability: public, product, WC & PI
CTP
We decided that the insurance company we are concerned with will have 5 lines of business. These are motor, household, fire&ISR, liability and CTP. These are the 5 largest lines of business in the Australian industry by earned premium and we chose these lines specifically to give a broad mix and realistic representation of the industry.

15. Numerical Simulation Correlation matrix input Line of Business Motor
Household
Fire & ISR
Liability
CTP
100%
20%
50%
10% 0%
25%
We also needed a linear correlation matrix input for our simulation.
We started off by inferring a matrix from the historical data between 1992 and However, we found that the levels of correlation that resulted were counter-intuitive in many cases, with some pair-wise correlations as high as 80% and CTP was negatively correlated with the other classes.
The matrix shown here, is the one we actually used in our simulation. This was derived with reference to the Tillinghast and Trowbridge reports from We emphasise that we have chosen positive values for all pair-wise correlations and that these values are in fact very subjective.

16. Numerical Simulation Marginal distribution input Line of business
Motor
Gamma
Household
Fire & ISR
Log-normal
Liability
CTP
We also need to input the marginal distributions for each business line’s loss. Here we chose our distributions from the Gamma and log-normal distributions because these are commonly used claim size distributions in general insurance. We basically matched the tail weight of the distributions to each line of business, with the lighter tailed classes being assigned a Gamma distribution, and the heavier tailed lines being assigned a log-normal distribution.
We estimate the parameters of each marginal based on the method of moments using the historic industry data from 1992 to 2002.

17. Results of Simulation Normal copula
For each copula, we generated 1000 simulations of each business line’s loss. To check for reasonableness of our simulation, we visually observe the pair-wise scatter plots of the simulated values.
Here we have the case of the normal copula. We see that the dependence structure is not very strong. In fact, there is only evidence of linear correlation with positive sloping patterns for some pairs of business lines and there is no evident tail dependence. These observations are reasonable given that the Normal copula allows only for linear dependence.

18. Results of Simulation Student-t (3 df) copula
Switching to the Student-t copula with 3 df, we see that the pair-wise relationships are no longer linear in nature. We see that there is evidence of tail dependence, especially in the upper tails of the loss distributions where a large loss from one line of business leads to a large loss in another line.

19. Results of Simulation Student-t (10 df) copula
If we increase the degrees of freedom to 10, we see a much more similar picture to the Normal copula. This is expected as the normal copula is really the asymptotic case of a Student-t copula as the degrees of freedom goes to infinity.

20. Results of Simulation Cauchy copula
For the Cauchy copula, we noted earlier that it allowed for the heaviest tail dependence of the copulas we consider. This is evident from this scatter plot.
Also, here we see some examples of non-linear dependence structures such as the quadratic relationship between household and fire&ISR losses.

21. Results of Simulation Independence copula
Finally, for the independence copula, we see that the losses are randomly scattered in the pair-wise plot, which is what we had expected.

22. Results of Simulation Aggregated loss, Z, under each copula
Next we aggregate the individual line’s losses by the industry earned premium as at June 2002.
Here we see the different aggregate loss distributions resulting from each of the copulas. This is the first indication of different capital requirements under each copula.
Mean – roughly the same
Dispersion – higher for more tail dependent copulas
Skewness – All are skewed with a positive tail – this is reasonable for insurance losses; again, tail dependent copulas induce a more skewed aggregate distribution.
We note the loss distribution is in terms of gross loss ratios, that is the implied capital requirements and diversification benefits will be quoted on a per unit of gross earned premium basis.

23. Results of Simulation Capital requirements (CRs)
Note: risk measures 1 – 4 are VaR(97.5%), VaR(99.5%),TCE(97.5%) and TCE(99.5%) respectively.
Note, risk measures 1 – 4 are VaR (97.5%), VaR(99.5%),TCE(97.5%) and TCE(99.5%) respectively

24. Results of Simulation Diversification benefits (DBs)
Note: risk measures 1 – 4 are VaR(97.5%), VaR(99.5%),TCE(97.5%) and TCE(99.5%) respectively.
Since we now have a distribution of simulated values of each line’s loss under each copula, we can readily work out the capital requirements if each line is treated as a stand alone business. Consequently, we can use this information to deduce the diversification benefit from operating multiple lines of business, i.e., to hold capital on the aggregate loss rather than holding the aggregate of the capital on each individual line’s loss.
Note, risk measures 1 – 4 are VaR (97.5%), VaR(99.5%),TCE(97.5%) and TCE(99.5%) respectively

25. Results of Simulation
Comparison with Prescribed Method (PM) – industry portfolio
Normal
t (3 df)
t (10 df)
Cauchy
Independence
PM CR
VaR 99.5% CR
Excess Capital
% Savings
7.84%
2.79%
6.51%
-2.35%
7.73%
PM is a formula based attempt at calculating the capital requirements based on the risks faced by an insurer. The calculation procedures is set out by APRA.
Comment on comparison

26. Results of Simulation
Comparison with Prescribed Method (PM) – short tail portfolio
Normal
t (3 df)
t (10 df)
Cauchy
Independence
PM CR
VaR 99.5% CR
Excess Capital
% Savings
7.85%
4.31%
6.89%
1.54%
19.45%
It is also worthwhile to compare the PM and copula capital requirements for arbitrarily chosen portfolios writing predominantly short tail and long tail business. Firstly, for a short tail portfolio, we find that regardless of the copula assumed, an insurer can make capital savings compared to using APRA’s prescribed method.
Comment on comparison

27. Results of Simulation
Comparison with Prescribed Method (PM) – long tail portfolio
Normal
t (3 df)
t (10 df)
Cauchy
Independence
PM CR
VaR 99.5% CR
Excess Capital
% Savings
7.00%
-3.46%
6.31%
-12.75%
-17.21%
The pattern is not as clear cut for the long tail portfolio.
These comparison leads to 2 issues:
The inconsistency between the results for the short and long tail portfolios highlights the deficiency of the PM as a “one size fits all” tool for determining the correct amount of capital.
There is an urgent need for more flexible internal models to be developed which incorporate an accurate treatment of the dependencies across business lines.
Comment on comparison

28. Key Findings
Choice of copula matters dramatically for both CRs and DBs
More tail dependent  higher CR
More tail dependent  higher DB
Need to select the correct copula for the insurer’s specific dependence structure
CR and DB shares a positive relationship
PM is not a “one size fits all” solution
Mis-estimations of the true capital requirement

29. Limitations Simplifying assumptions Underwriting risk only
Ignores impact of reinsurance
Ignores impact of investment
Results do not quantify the amount of capital required
Comparison between copulas
Not comparable with results of other studies

30. Further Research Other copulas Isaacs (2003) used the Gumbel
Other types of risk dependencies
E.g., between investment and operational risks
Relax some assumptions
Include reinsurance
Factor in expenses
Factor in investments